ON THE CONDUCTION OF HEAT IN CRYSTALS. 225 are proportional to A~lx, B~ly, 0~lz. Hence the differential equations of a line of motion are _ dx_ _ _ _ dy x - Taking £ 97, f, to denote the same quantities as in Art. 10, and putting for shortness we get - a>*7 + G 77 - a> + a> - Conceive an elastic solid to be fixed at the origin, and to expand alike in all directions and at all points with a velocity of expansion unity, so that a particle which at the end of the time t is situated at a distance r from the origin, at the end of the time t + dt is situated at a distance r(b+d£). Conceive this solid at the same time to turn, with an angular velocity oo equal to V(®/2+6)//2 + o>w2), about an axis whose direction-cosines are a/aT1, a/V"1, ©'"co"1. The direction of motion of any particle will represent the direction of the flow of heat in what we may still call the auxiliary solid, from whence the direction of the flow of heat in the given solid will be obtained by merely conceiving the whole figure differently magnified or diminished in three rectangular directions. This rotatory sort of motion of heat, produced by the mere diffusion from the source outwards, certainly seems very strange, and leads us to think, independently of the theory of molecular radiation, that the expressions for the flux with six arbitrary constants only, namely the expressions (8), or the equivalent expressions (7), are the most general possible. 20. Let the auxiliary solid be referred to the rectangular axes of f ', 77', §", of which the last coincides with the axis to which co refers. It may be seen immediately, without analytical transformation, that the differential equations to the lines of motion will be dp drf _d? - - CM S. ITT. 15